Six-loop $\varepsilon$ expansion study of three-dimensional $O(n)\times O(m)$ spin models

TL;DR

This study performs a six-loop epsilon expansion analysis of three-dimensional $O(n) imes O(m)$ spin models, providing detailed critical behavior insights for frustrated spin systems with complex ordering.

Contribution

It extends epsilon expansions up to fifth order for these models and constructs stability diagrams, offering refined estimates of critical exponents and phase transition nature.

Findings

Confirmation that certain cases undergo first-order phase transitions.

Provision of explicit epsilon series for various m values.

Construction of stability diagrams for fixed points.

Abstract

The Landau-Wilson field theory with $O(n)\times O(m)$ symmetry which describes the critical thermodynamics of frustrated spin systems with noncollinear and noncoplanar ordering is analyzed in $4 - \varepsilon$ dimensions within the minimal subtraction scheme in the six-loop approximation. The $\varepsilon$ expansions for marginal dimensionalities of the order parameter $n^H(m,4-\varepsilon)$, $n^-(m,4-\varepsilon)$, $n^+(m,4-\varepsilon)$ separating different regimes of critical behavior are extended up to $\varepsilon^5$ terms. Concrete series with coefficients in decimals are presented for $m=\{2, \dots, 6\}$. The \textit{diagram of stability} of nontrivial fixed points, including the chiral one, in $(m,n)$ plane is constructed by means of summing up of corresponding $\varepsilon$ expansions using various resummation techniques. Numerical estimates of the chiral critical exponents for several couples $\{m,n\}$ are also found. Comparative analysis of our results with their counterparts obtained earlier within the lower-order approximations and by means of alternative approaches is performed. It is confirmed, in particular, that in physically interesting cases $n=2, m=2$ and $n=2, m=3$ phase transitions into chiral phases should be first-order.